---
title: "Stability of h(t) = e^(4t)u(t)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:stability-e4t-ut"
source: https://www.bctengineer.com/q/semester_5/data_communication/stability-e4t-ut
site: BCT Engineering
---

# Stability of h(t) = e^(4t)u(t)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2078 Kartik, 4 marks
- Video: [Stable and Unstable Systems](https://www.youtube.com/watch?v=ngJ2QWvMPbI) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/stability-e4t-ut
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Test the stability of the system h(t) = e^(4t)·u(t).

## Short answer

The system is unstable because its impulse response is not absolutely integrable: ∫0^∞ e^{4t} dt = ∞.

## Answer

An LTI system is BIBO stable if its impulse response is absolutely integrable: ∫|h(t)| dt < ∞ over (-∞, ∞).

∫|h(t)| dt = ∫₀^∞ e^(4t) dt = [e^(4t)/4]₀^∞ = ∞

The integral diverges because the exponent is positive (a pole in the right half of the s-plane at s = 4).

**Answer: the system is unstable.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. BIBO stable if
   A. h = 0
   B. integral |h| finite
   C. h(t) > 0
   D. h causal
   Answer: B. Absolutely integrable. (Key point: Condition)

2. e^(4t)u(t) is
   A. stable
   B. marginal
   C. unstable
   D. anticausal
   Answer: C. Integral diverges. (Key point: Result)

3. Pole at s = 4 is in
   A. left half-plane
   B. origin
   C. jw axis
   D. right half-plane
   Answer: D. Positive exponent. (Key point: Pole)

## Related in Data Communication

- [Stable/unstable systems; test stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-define-test-72c) — Data Communication, chapter 2
- [Stability of h(t) = t·e^(-3t)u(t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-te-3t) — Data Communication, chapter 2
- [Recursive/nonrecursive systems; stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Arecursive-nonrecursive-stability) — Data Communication, chapter 2
- [Range of a, b for stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-range-ab) — Data Communication, chapter 2
- [Stability of y(t) = x(t²)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-xt2) — Data Communication, chapter 2

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